Short Combinatorial Proof that the DFJ Polytope is contained in the MTZ Polytope for the Asymmetric Traveling Salesman Problem

TL;DR

This paper presents a concise combinatorial proof demonstrating that the DFJ polytope is contained within the MTZ polytope for the ATSP, simplifying previous lengthy analytic proofs by using graph distance relations.

Contribution

It introduces a shorter, combinatorial proof of the containment relation between the DFJ and MTZ polytopes for the ATSP.

Findings

The proof confirms the polytope containment.

The combinatorial approach simplifies understanding of the relation.

The method relates polytope properties to graph distances.

Abstract

For the Asymmetric Traveling Salesman Problem (ATSP), it is known that the Dantzig-Fulkerson-Johnson (DFJ) polytope is contained in the Miller-Tucker-Zemlin (MTZ) polytope. The analytic proofs of this fact are quite long. Here, we present a proof which is combinatorial and significantly shorter by relating the formulation to distances in a modified graph.